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NASA Technical Memorandum 84281

A Mathematical Model of a Single Main Rotor Helicopter for Piloted Simulation Peter D. Talbot, Bruce E. Tinling, William A. Decker, and Robert T. N. Chen

t

September 1982

NASA National Aeronautics and Space Administration

...

https://ntrs.nasa.gov/search.jsp?R=19830001781 2018-05-26T14:14:39+00:00Z

NASA Technical Memorandum 84283

A Mathematical Model of a Single Main Rotor Helicopter for Piloted Simulation Peter D. Talbot Bruce E. Tinling William A. Decker Robert T. N. Chen, Ames Research Center, Moffett,Field, California

National Aeronautics and Space Administration

Ames Research Center Moffett Field California 94035

CONTENTS

Page

SuMMAEtY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I .

SIMULATION MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main r o t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 T a i l r o t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empennage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuselage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor R o t a t i o n a l Degree of Freedom and RPM Governing . . . . . . . . . . Contro l Systems

Linear ized Six-Degree-of-Freedom Model . . . . . . . . . . . . . . . . .

r I

c, . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDICES A Nota t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B A x i s s y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Main Rotor Flapping Dynamics and Force and Moment Ca lcu la t ion . . . . D T a i l Rotor Flapping and Force Calcu la t ion . . . . . . . . . . . . . . E Empennage Forces and Moments . . . . . . . . . . . . . . . . . . . . F Ca lcu la t ion of Fuselage Forces and Moments . . . . . . . . . . . . . G RPM Governor . . . . . . . . . . . . . . . . . . . . . . . . . . . . H Cockpit Con t ro l s and Cycl ic Control Phasing . . . . . . . . . . . . . I Linea r i zed Six-Degree-of-Freedom Representa t ion of He l i cop te r

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J Conf igura t ion Desc r ip t ion Requirements . . . . . . . . . . . . . . .

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

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13

15

23

28

32

35

37

39

4 1

46

iii

k

TABLES

Page

1-1 Elements of the Linearized Equations of Motion . . . . . . . . . . . . . 40

I J-1 Configuration Description Requirements . . . . . . . . . . . . . . . . . 42

V

FIGURES

Page

1

2

3

4

B- 1

B-2

D- 1

Block Diagram Showing P r i n c i p a l Elements of S ing le Rotor He l i cop te r M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . Typical Var i a t ion of Empennage L i f t and Drag C o e f f i c i e n t s

Block Diagram of RPM Governor

S t r u c t u r e of Control System Model . . . . . . . . . . . . . . . . . . .

Components Defined . . . . . . . . . . . . . . . . . . . . . . . . . Hub-Body, A i r c r a f t Reference and Body-c.g. Axis Systems . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

The Hub-Wind Axis System wi th Main Rotor Force, Moment, and Veloc i ty

T a i l Rotor Forces and Moments . . . . . . . . . . . . . . . . . . . . .

5

6

7 3

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24

v i i

A MATHEMATICAL MODEL OF A SINGLE MAIN ROTOR HELICOPTER

FOR PILOTED SIMULATION

P e t e r D. Ta lbot , Bruce E. T in l ing , W i l l i a m A. Decker, and Robert T. N . Chen

Ames Research Center

SUMMARY

4 This r e p o r t documents a h e l i c o p t e r mathematical model s u i t a b l e f o r p i l o t e d

s imula t ion of f l y i n g q u a l i t i e s . The mathematical model i s a non l inea r , t o t a l f o r c e and moment model of a s i n g l e main r o t o r h e l i c o p t e r . freedom: s i x rigid-body, t h r e e ro tor - f lapping , and the r o t o r r o t a t i o n a l degrees of freedom. i n t e g r a t e d and summed about t h e azimuth. d e t a i l e d r e p r e s e n t a t i o n over a nominal angle of a t t a c k and s i d e s l i p range of ?15", and i t u s e s a s i m p l i f i e d curve f i t a t l a r g e ang le s of a t t a c k o r s i d e s l i p . S t a b i l i z - i n g s u r f a c e aerodynamics are modeled with a l i f t curve s lope between s t a l l l i m i t s and a gene ra l curve f i t f o r l a r g e ang le s of a t t a c k . A genera l ized s t a b i l i t y and c o n t r o l augmentation system i s descr ibed . Addi t iona l computer sub rou t ines provide op t ions f o r a s i m p l i f i e d engine/governor model, atmospheric turbulence, and a l i n e a r i z e d six-degree-of-freedom dynamic model f o r s t a b i l i t y and c o n t r o l a n a l y s i s .

me model h a s t en degrees of

The r o t o r model assumes r i g i d b l ades wi th r o t o r f o r c e s and moments r a d i a l l y The f u s e l a g e aerodynamic model u ses a

I N TROD UC T I ON

An expanded f l y i n g - q u a l i t i e s da t a base i s needed f o r u se i n developing des ign c r i t e r i a f o r f u t u r e h e l i c o p t e r s . A safe and c o s t - e f f e c t i v e way t o e s t a b l i s h such a d a t a base i s t o conduct explora tory i n v e s t i g a t i o n s us ing p i l o t e d ground-based simu- l a t o r s , and then t o s u b s t a n t i a t e t h e r e s u l t s i n f l i g h t u s ing v a r i a b l e s t ab i l i t y research h e l i c o p t e r s .

A mathematical model s u i t a b l e f o r real-time p i l o t e d s imula t ion of s i n g l e r o t o r h e l i c o p t e r s h a s been implemented a t Ames Research Center. A s descr ibed i n r e f e r - ence 1, s imula t ion models used a t Ames Research Center c o n s i s t of a common c o r e of rigid-body equat ions and an aerodynamic model t h a t provides the aerodynamic f o r c e s and moments. This r e p o r t documents the equa t ions used i n t h e aerodynamic model.

The r e p o r t c o n s i s t s of a b r i e f d e s c r i p t i o n of t h e o v e r a l l model and i t s compo- n e n t s and appendices t h a t d e t a i l t h e equat ions used i n the model and t h e parameters r equ i r ed t o desc r ibe a h e l i c o p t e r conf igura t ion .

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SIMULATION MODEL

The o v e r a l l arrangement of t he s imula t ion model i s shown i n f i g u r e 1. The p r i n c i p a l assumptions and cons ide ra t ions employed i n developing each element of t h e model are given i n t h e main body of the r e p o r t ; d e t a i l e d equat ions f o r t h e f o r c e s and moments are given i n the appendices. The model e lements , denoted T i i n f i g - u r e 1, are requ i r ed t o achieve t r a n s f e r of v e l o c i t i e s , f o r c e s , and moments from

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one a x i s system t o another ; i n some ins t ances , t o account f o r aerodynamic i n t e r f e r - ence e f f e c t s between model components. d i c e s where r equ i r ed .

i These elements are descr ibed i n t h e appen- I

The n o t a t i o n employed is l i s t e d i n appendix A, and a d e s c r i p t i o n of t h e va r ious a x i s systems is given i n appendix B. t h e computer program have been chosen t o be easi ly i d e n t i f i a b l e from t h e n o t a t i o n used he re in . f i e d i n the computer program l i s t i n g s .

The v a r i a b l e names i n t h e FORTRAN coding f o r

I Through t h i s mnemonic dzvice, equat ions I n the appendix can be i d e n t i -

?A Main Rotor

The development of t he equa t ions desc r ib ing t h e dynamics and t h e f o r c e s and r, moments a c t i n g on t h e main r o t o r are given i n d e t a i l i n r e f e r e n c e s 2 and 3 . This

mathematical r e p r e s e n t a t i o n e x p l i c i t l y accounts f o r t h e dynamic e f f e c t of r o t o r modes, such as ro tor -b lade f l app ing , which can be i n a frequency range which i s important i n s t u d i e s of f l y i n g q u a l i t i e s . For t h e r o t o r model descr ibed i n t h i s r e p o r t , t he f l a p - p ing dynamics were approximated us ing a t ip -pa th p lane r e p r e s e n t a t i o n .

The f l a p p i n g equat ion of motion of t h e r o t o r b l ade w a s f i r s t developed us ing t h e fo l lowing assumptions. The assumptions are similar t o those used f o r t h e "classicall' equa t ions ( r e f s . 4 and 5).

1. The r o t o r b l ade w a s r i g i d i n bending and t o r s i o n , and the t w i s t of t he b l a d e w a s l i n e a r .

2. The f l a p p i n g angle and inf low ang le were assumed t o be s m a l l and t h e analy- s is u t i l i z e d a simple s t r i p theory.

3. The e f f e c t s of t he a i rcraf t motion on t h e b l ade f l a p p i n g were l i m i t e d t o those due t o t h e angular a c c e l e r a t i o n normal a c c e l e r a t i o n .

and 4, t h e angular ra te p and q , and t h e

4 . The reversed flow reg ion w a s ignored and t h e compress ib i l i t y and s ta l l e f f e c t s d i s r ega rded .

5 . The inf low w a s assumed t o be uniform and no inf low dynamics were used.

6 . The t i p - l o s s f a c t o r was assumed t o be 1.

The f l a p p i n g equat ions of motion e x p l i c i t l y con ta in t h e primary design param-

> f l a p coupl ing . The b lade f l a p p i n g i n those equat ions w a s then approximated by t h e eters, namely: f l a p p i n g h inge r e s t r a i n t , h inge o f f s e t , b l a d e Lock number, and p i t c h -

f i r s t harmonic terms wi th time-varying c o e f f i c i e n t s , t h a t i s ,

7 . B(t) = a o ( t ) - a l ( t ) c o s $ - b l ( t ) s i n $.

In t h e developmenk of t h e equat ions f o r f o r c e s and moments, t h e same se t of basic a s s u m p t i m s ( 1 through 7 above) , discussed i n conjunct ion w i t h the development of t h e t i p -pa th p lane dynamic equat ions , w a s u t i l i z e d . 'rhus, aerc,dynamically, momentum theory w a s used i n conjunct ion w i t h t h e uniform inf low; s i m p l e s t r i p theory w a s u t i i i z e d and the b l ade f o r c e s were a n a l y t i c a l l y integrate .1 over the r ad ius . Because t h e reversed flow reg ion and the s t a l l and compress ib i l i t y e f f e c t s w e r e ignored , t h e t o t a l r o t o r f o r c e s and moments were aga in ana ly t i caL ly obta ined by

3

summing t h e c o n t r i b u t i o n s , t o each b lade , that w e r e a n a l y t i c a l func t ions of t h e azimuth. sis are v a l i d only f o r a l i m i t e d range of f l i g h t cond i t ions . ous s tudy ( r e f . 5) has shown t h a t t h i s type of a n a l y s i s i s v a l i d f o r s t a b i l i t y and c o n t r o l i n v e s t i g a t i o n s of t he r o t o r c r a f t up t o an advance r a t i o of about 0.3. similar t o t h e development of t h e t ip -pa th p l ane dynamic equat ions , t h e s e r o t o r f o r c e s and moments were f i r s t obtained i n t h e wind-hub coord ina te system. then transformed i n t o t h e hub-body system (see appendix C ) .

Because of t h e s e assumptions and s i m p l i f i c a t i o n s , t h e r e s u l t s of the analy- Never the less , a previ -

Also,

They were

The f o r c e s and moments thus developed con ta in p e r i o d i c terms; t h e h i g h e s t har- monic terms correspond d i r e c t l y t o t h e number of r o t o r b lades . For example, f o r a three-bladed r o t o r , t he f o r c e and moment equa t ions con ta in only t h r e e / r e v o l u t i o n harmonic terms, and f o r a four-bladed r o t o r , f o u r / r e v o l u t i o n harmonic terms. The frequency of t hese harmonic terms i s s u f f i c i e n t l y h igh t o b e of no i n t e r e s t t o han- d l i n g q u a l i t y i n v e s t i g a t i o n s . These t e r m s have t h e r e f o r e been de le t ed . The r e s u l t - -' i n g f o r c e and moment express ions are given i n appendix C.

3

A development t o modify these f o r c e s and moment equa t ions t o inc lude t h e e f f e c t s of nonuniform inf low, similar t o t h e development f o r t h e f l a p p i n g equa t ions i n r e f - e rences 6 and 7 , i s i n progress . These equat ions , a long wi th t h e modified t ip -pa th p lane r e p r e s e n t a t i o n given i n r e fe rence 7 w i l l la ter supersede those shown i n appendix C.

T a i l Rotor

The t a i l r o t o r w a s modeled as a t e e t e r i n g r o t o r wi thout c y c l i c p i t c h . For t h i s ca se , t he f o r c e s i n the wind-hub system may be obta ined from the express ions de r ived f o r t h e main r o t o r by s e t t i n g t h e l a te ra l and l o n g i t u d i n a l c y c l i c p i t c h terms (Alc

and Blc) equa l t o zero.

h igher than t h a t of t he main r o t o r system, t h e t ip -pa th p lane dynamics may b e neglec ted . Thus, f o r t a i l r o t o r a p p l i c a t i o n s , the f i r s t and second d e r i v a t i v e s of t h e blade-f lapping nonro ta t ing coord ina te s .. are set equal t o ze ro i n t h e f o r c e equa- t i o n s (io = i 1 = 61 = 0 s t a t i c f o r c e express ions s imilar t o those i n c lass ical work ( r e f s . 3 and 4 ) .

Fur the r , s i n c e the t a i l r o t o r f l a p p i n g frequency is much

and 20 = 81 = b l = 0 ) . The r e s u l t i s a set of b a s i c quasi-

The l o c a l flow a t t h e t a i l r o t o r i nc ludes t h e e f f e c t of downwash from t h e main r o t o r system. t a i l r o t o r f o r c e s are given i n appendix D.

The method employed t o e s t i m a t e t h i s downwash and t h e equa t ions f o r

Empennage -3

The l i f t and drag f o r c e s on t h e ver t ica l f i n and h o r i z o n t a l t a i l are approxi- mated f o r a l l angle of a t t a c k and s i d e s l i p , i nc lud ing rearward f l i g h t . made f o r t h e a d d i t i o n of terms due t o main rotor- induced v e l o c i t i e s a t t h e h o r i z o n t a l -: t a i l , and t a i l r o t o r v e l o c i t i e s a t t h e v e r t i c a l f i n .

P rov i s ion i s

The p r i n c i p a l assumptions made i n developing t h e expres s ions f o r t h e f o r c e s and moments due t o the ver t ica l f i n and h o r i z o n t a l t a i l are as fo l lows:

1. The l i f t and drag f o r c e s are app l i ed a t t h e q u a r t e r chord of each s u r f a c e a t t he spanwise l o c a t i o n of t h e c e n t e r of area.

4

2.

3.

The a i r f o i l p r o f i l e s are symmetrical.

The l i f t curve s lope p r i o r t o s t a l l i s given by simple l i f t i n g - l i n e theory I

assuming an e l l i p t i c a l l i f t d i s t r i b u t i o n w i t h uniform downwash. appl ied f o r s i d e s l i p and f o r sweep of t h e v e r t i c a l f i n .

Cor rec t ions are I- -

4 . Maximum l i f t c o e f f i c i e n t i s spec i f i ed ; however, i f t h e l i f t curve s lope is i i s assumed such t h a t t h i s va lue i s n o t reached a t a n ang le of attack of ~ / 4 , C b a x

~

I t o occur a t t h i s ang le of attack.

c 5. Pos t s t a l l v a r i a t i o n of l i f t c o e f f i c i e n t i s based on C b a x decreas ing by

20% as t h e ang le of a t t a c k is increased by 20%, and fo l lowing a p a r t i c u l a r v a r i a t i o n t h e r e a f t e r t o reach zero l i f t a t an angle of a t t a c k of . ~ / 2 .

6 . L i f t c o e f f i c i e n t i n rearward f l i g h t is 80% of t h a t i n forward f l i g h t .

7. The p r o f i l e drag c o e f f i c i e n t v a r i e s w i t h angle of a t t a c k and reaches a va lue of 1 when a = 27~12 .

8. c o e f f i c i e n t .

The induced drag c o e f f i c i e n t v a r i e s as t h e square of t h e c a l c u l a t e d l i f t

A t y p i c a l v a r i a t i o n of t he l i f t and drag c o e f f i c i e n t s f o r an empennage su r face i s shown i n f i g u r e 2. r equ i r ed t ransformat ions f o r v e l o c i t i e s and f o r c e s are given i n appendix E.

Expressions f o r c a l c u l a t i n g the empennage f o r c e s and the

Figure 2 . - Typical v a r i a t i o n of empennage l i f t and drag c o e f f i c i e n t s .

Fuselage

The aerodynamic model of t h e fuse lage must f u l f i l l two requirements. The f i r s t i s t o provide an e s t ima te of t he fo rces and moments a t s m a l l ang le s of a t t a c k and

5

s i d e s l i p t h a t w i l l b e encountered a t s u b s t a n t i a l forward speeds. r e p r e s e n t a t i o n of t h e important e f f e c t s of fu se l age aerodynamics on performance and s t a b i l i t y a t t h e s e speeds. t i o n of f o r c e s and moments over t h e e n t i r e range of ang le of at tack and s i d e s l i p (0" t o +180°) t h a t can be encountered i n approach t o hovering f l i g h t o r i n hover. Cont inui ty i s requi red t o avoid sudden u n r e a l i s t i c l i n e a r o r angu la r a c c e l e r a t i o n s i n response t o a small change i n a t t i t u d e . Accuracy of t h e model a t extreme a t t i - tudes is considered t o be of secondary importance, s i n c e t h e f u s e l a g e f o r c e s a t these speeds are very s m a l l compared t o t h e r o t o r fo rces .

Th i s provides a

The second requirement is t o provide a cont inuous varia-

A technique has been developed t o provide a cont inuous model by f i t t i n g typ i - ?

tal v a r i a t i o n s of the f o r c e s and moments through d a t a p o i n t s ob ta ined a t s p e c i f i c widely sepa ra t ed angles of a t t a c k and s i d e s l i p i n a wind tunnel ( s ee r e f . 8 ) . How- eve r , even t h i s spa r se l e v e l of test d a t a f o r t h e fuse l age is t y p i c a l l y unava i l ab le and an a l t e r n a t i v e technique must b e employed.

The model employed he re in relies on s e p a r a t e r e p r e s e n t a t i o n s f o r ang le s of a t t a c k and s i d e s l i p i n t h e range from -15" t o 15' and from 230" t o +180". is provided by a l i n e a r i n t e r p o l a t i o n f o r f o r c e s and moments i n the ang le range n o t covered. The f o r c e s and moments f o r t h e lower ang le range are obta ined from test da ta o r from estimates based on d a t a from similar fuse lages . The d a t a f o r t h e h igh ang le of a t t a c k and s i d e s l i p range are based on the es t imated magnitude and l o c a t i o n of t h e drag f o r c e vec to r when t h e f u s e l a g e i s i n a 90" c r o s s f low, and on an approxi- mation t o i t s observed v a r i a t i o n s wi th a t t i t u d e from wind-tunnel tests of bod ie s of r evo lu t ion .

Cont inui ty

Details of the procedure f o r e s t ima t ing fuse l age f o r c e s and moments are given i n appendix F.

Rotor Ro ta t iona l Degree of Freedom and RPM Governing

An opt ion i s a v a i l a b l e which provides f o r a r o t a t i o n a l degree of freedom f o r t h e r o t o r . When t h i s opt ion i s used, t he main r o t o r and the t a i l r o t o r r o t a t i o n a l speeds vary according t o the c u r r e n t torque requirements and t h e engine power a v a i l a b l e . The i n i t i a l trim cond i t ions e s t a b l i s h b a s e l i n e v a l u e s of r o t o r speeds and engine torque. Deviat ions from t h e s e b a s e l i n e va lues change the r o t o r t o rque requirements and, hence, r o t o r speed . These changes i n speed cause t h e rpm governor t o vary t h e f u e l f low t o the engine t o change the power t o main ta in t h e d e s i r e d angu la r rate. A block diagram of t h e rpm governor i s shown i n f i g u r e 3 . dynamic model f o r t h i s degree of freedom and t h e rpm governor are given i n

Fur the r d e t a i l s of the

/ (MAIN AND TAIL ROTOR 4 Q ~ o appendix G.

THROTTLE TORQUE REQUIRED) (0 N/O F F )

A W V AHP 550 wf An -Ai2

* KE -m - b s2 4

GOVERNOR

As2 %ET G (SI =a,

s1 + - Figure 3 . - Block diagram of rpm governor.

6

Control Systems

4

~ f

c-

3 .

The h e l i c o p t e r model has a genera l ized c o n t r o l system t h a t a c c e p t s i npu t s from t h e p i l o t , f a c i l i t a t e s c o n t r o l augmentation and s t a b i l i t y augmentation, and provides f o r mechanical c o n t r o l mixing o r phasing of the c y c l i c i n p u t s . t h i s system i s shown i n f i g u r e 4 .

A block diagram of The c o n t r o l augmentation system employs a

*

6 6ap -

C p j K8 XO 6

13 4 K M 3 b DIRECTIONAL + eP

CYCLIC , CONTROL +

LATERAL CYCLIC CONTROL PHASING -,AIRCRAFT DYNAMICS LONGITUDINAL CYCLIC CONTROL AND

COLLECTIVE CONTROL GEARING AIRCRAFT *

AND + STATE, X RIGGING

XO

X O - T CROSS-FEED AND

FEED FORWARD GAINS FEEDBACK GAINS X = (u, w, 0 , 8 , v, p, 4, r )

Figure 4 . - St ruc tu re of c o n t r o l system model.

s t r u c t u r e t h a t p e r m i t s implementation of c ros s feeds from each of t he four cockpi t c o n t r o l s ; namely, l o n g i t u d i n a l and l a t e ra l c y c l i c s t i c k , c o l l e c t i v e s t i c k , and d i r e c t i o n a l peda ls . The feedback ga in s t r u c t u r e of the s t a b i l i t y augmentation system permi ts feedback p ropor t iona l t o any element of the s t a t e vec to r t o each of t h e f o u r c o n t r o l s . The e n t i r e c o n t r o l s t r u c t u r e a l s o f a c i l i t a t e s ga in schedul ing as f u n c t i o n s of t h e f l i g h t parameters , such a s a i r speed .

Details concerning the ze ro pos i t i on and s ign convent ion of t he cockpi t c o n t r o l s and t h e mechanical c y c l i c c o n t r o l phasing l o g i c are given i n appendix H.

Atmospheric Turbulence

The r e p r e s e n t a t i o n of atmospheric tu rbulence i s based on t h e Dryden model and is desc r ibed i n MIL-F-8785B ( r e f . 9 ) . The i n p u t s requi red f o r t h i s model are t h e a i r - c r a f t v e l o c i t y r e l a t i v e t o the a i r mass, t he turbulence scale l eng ths , and the r m s

7

g u s t l e v e l s . s p e c i f i e d as func t ions of a l t i t u d e . v e l o c i t y .

For t h i s r ep resen ta t ion , scale l eng th and t h e wind v e l o c i t y can b e The rms g u s t l e v e l s are dependent upon wind

Linear ized Six-Degree-of-Freedom Model

A computer subrout ine i s a v a i l a b l e t h a t genera tes t h e c o e f f i c i e n t s of a l i n e a r , f i r s t - o r d e r set of d i f f e r e n t i a l equat ions t h a t r e p r e s e n t s t h e r i g i d body dynamics of t h e h e l i c o p t e r f o r small p e r t u r b a t i o n s from a f i x e d ope ra t ing po in t . assumption necessary t o genera te t h i s l i n e a r se t of equat ions is that t h e h e l i c o p t e r i n i t i a l angular rates are zero. The d i f f e r e n t i a l equat ions are of t h e form: G

The p r i n c i p a l

k = [FIX + [ G I 6

where x r ep resen t s p e r t u r b a t i o n s from t r i m of the state v a r i a b l e s u , w, q , 8 , v , p , $, and r ; and 6 r e p r e s e n t s the c o c t r o l displacements from t r i m A s e , A6,, Asa, and A $ . i s given i n appendix I.

The genera t ion of t he F and G matrices and a d e s c r i p t i o n of each element

8

APPENDIX A

NOTATION

blade l i f t - c u r v e s lope , p e r rad a

b lade coning ang le measured from hub p lane i n t h e hub-wind axes system, r ad

l o n g i t u d i n a l f i r s t -harmonic f lapping c o e f f i c i e n t measured from t h e hub p lane i n t h e wind-hub axes system, r a d

l o n g i t u d i n a l f i r s t -harmonic f lapping c o e f f i c i e n t measured from t h e hub p lane i n the hub-body axes system, r a d

la teral c y c l i c p i t c h measured from hub p lane i n t h e wind-wub axes system, rad

la teral c y c l i c p i t c h measured from hub p lane i n t h e hub-body axes system, rad

a spec t ra t i o , span2 /area AR

b l l a te ra l f i r s t -harmonic f lapping c o e f f i c i e n t measured from hub p lane i n t h e wind-hub axes system, rad

bh lateral f i rs t -harmonic f lapping c o e f f i c i e n t measured from hub p lane i n t h e hub-body axes system, r ad

BL l a t e r a l d i s t a n c e ( b u t t l i n e ) i n t h e fuse l age axes system, m ( f t )

l o n g i t u d i n a l c y c l i c p i t c h measured from hub p lane i n the wind-hub axes system, r ad

B I S l o n g i t u d i n a l c y c l i c p i t c h measured from hub p lane i n the hub-body axes system, rad

b l ade chord, m ( f t ) C

c o n t r o l gear ing c o n s t a n t s , n = 1 t o 8 cn

CB, C A I s )

t

c y c l i c c o n t r o l r i g g i n g cons t an t s , rad

drag c o e f f i c i e n t

CL l i f t c o e f f i c i e n t

maximum l i f t c o e f f i c i e n t

r o t o r t h r u s t c o e f f i c i e n t , T/p (,rrR2) (RR) CT

D drag f o r c e , N ( l b )

9

damping mat r ix i n f l a p p i n g d i f f e r e n t i a l equat ions

f l app ing h inge o f f s e t , m( f t )

g r a v i t a t i o n a l a c c e l e r a t i o n , m/sec2 ( f t / s e c 2 )

r o t o r fo rce normal t o s h a f t , p o s i t i v e downwind, N ( lb )

incidence of ver t ical f i n , p o s i t i v e f o r l ead ing edge t o t h e l e f t , r ad

incidence of h o r i z o n t a l s t a b i l i z e r , p o s i t i v e f o r l ead ing edge up, r ad

forward t i l t of r o t o r s h a f t w . r . t . f u s e l a g e , p o s i t i v e forward, r ad

r o t o r blade moment of i n e r t i a about f l app ing h inge , kg-m2 ( s lug - f t2 )

r o t o r moment of i n e r t i a about s h a f t

f a c t o r t o account f o r f r a c t i o n of v e r t i c a l t a i l i n t a i l r o t o r wake

f l app ing s p r i n g cons t an t , N-m/rad ( lb - f t / r ad )

p i tch- f lap coupl ing r a t i o , 4 t an 6,

s p r i n g matr ix i n f l a p p i n g d i f f e r e n t i a l equa t ions

fuse l age r o l l i n g moment, n-m (f t - lb )

fu se l age l i f t , N (lb)

r o l l i n g moment, p i t c h i n g moment, and yawing moment, r e s p e c t i v e l y , N-m ( f t - l b )

r o t o r blade mass, kg ( s lugs )

b l ade weight moment about f l a p p i n g h inge , N-m ( l b - f t )

number of b l ades

r o l l , p i t c h , and yaw rates i n t h e body-c.g. axes system, r ad / sec

r o l l , p i t c h , and yaw rates i n t h e body-c.g. axes system r e l a t i v e t o t h e a i r mass, rad /sec a .

r o l l , p i t c h , and yaw rates i n t h e hub-body axes system, r a d / s e c

r a t i o of f l app ing frequency t o r o t o r system angu la r v e l o c i t y

1 dynamic pressure , - p V 2 , N/m2 ( l b / f t 2 ) 2

10

Q R

S

S

STA

T s

W %

H W

vi

W f

WL

Z

a

%

6"

Y

6

P 6a

4. &C

6P

E

n

torque, N-m ( f t - l b )

r o t o r r a d i u s , m ( f t )

Laplace v a r i a b l e

area of s t a b i l i z i n g su r face , m2 ( f t 2 )

l o n g i t u d i n a l l o c a t i o n i n the fuse l age axes system, m ( f t )

t h r u s t , N ( l b )

l o n g i t u d i n a l , l a t e ra l , and ver t ical v e l o c i t i e s i n t h e body-c.g. system of axes , m/sec ( f t / s e c )

l o n g i t u d i n a l , l a t e ra l , and ver t ical v e l o c i t i e s r e l a t i v e t o t h e a i r mass i n t h e body-c.g. system of axes , m / s e c ( f t / s e c )

l o n g i t u d i n a l , l a t e ra l , and v e r t i c a l v e l o c i t i e s r e l a t i v e t o the a i r mass i n the a i r mass i n t h e hub-body system of axes , m/sec ( f t / s e c )

main r o t o r induced v e l o c i t y a t r o t o r d i s k , m/sec ( f t / s e c )

f u e l f low rate, N/hr ( l b / h r )

ver t ica l l o c a t i o n i n the fuse l age axes system, m ( f t )

l o n g i t u d i n a l , l a t e r a l , and ver t ical f o r c e s i n t h e body-c.g. axes system, N ( Ib )

s t a b i l i z i n g su r face ang le of a t t a c k , rad

a n g l e of a t t a c k f o r s t a l l , r a d

r o t o r s i d e s l i p ang le , r ad

b l ade Lock number, pacR4/Ig

equ iva len t r o t o r b lade p r o f i l e drag c o e f f i c i e n t

l a t e r a l c y c l i c s t i c k movement, p o s i t i v e t o r i g h t , c m ( in . )

c o l l e c t i v e c o n t r o l i n p u t , p o s i t i v e up, c m ( i n . )

l o n g i t u d i n a l c y c l i c s t i c k movement, p o s i t i v e a f t , c m ( in . )

peda l movement, p o s i t i v e r i g h t , c m ( in . )

h inge o f f s e t r a t i o , e / R

s t a b i l i z i n g s u r f a c e l i f t e f f i c i e n c y f a c t o r

11

0 Euler p i t c h angle , rad

00 b lade root c o l l e c t i v e p i t c h , rad

e t t o t a l blade t w i s t ( roo t minus t i p inc idence ) , rad

A WH CT in f low r a t i o A - -

= 2 ( v 2 + A 2 ) 1 / 2

A sweepback ang le of f i n , rad

/uH2 + V H 2

a R r o t o r advance r a t i o ,

P a i r d e s n t i y , kg/m3 ( s l u g s / f t 3 >

(5

T t i m e cons tan t

9 Euler r o l l ang le , rad

R r o t o r angular v e l o c i t y , rad /sec

Sub scr ipts :

r o t o r s o l i d i t y r a t i o , b l ade a r e a / d i s k a rea

B

c .g .

E

f

F

H

HS

i

MR

P

t

TR

W

body-c.g. axes system relat ive t o a i r mass

c e n t e r of g r a v i t y

engine

fuse l age

v e r t i c a l f i n

hub-body axes s y s t e m , hub l o c a t i o n

horizon t a l s t a b i l i z e r

induced

main r o t o r

p i l o t input

t h r o t t l e

t a i l r o t o r

hub-wind system of axes

1 2

APPENDIX B

AXIS SYSTEMS

Hub Wind

The hub-wind a x i s system is used i n t h e c a l c u l a t i o n of r o t o r f o r c e s and moments. The o r i g i n of t h e system i s t h e r o t o r hub, and t h e t h e r o t o r s h a f t . t i ve wind normal t o t h e s h a f t a x i s , and t h e right-handed or thogonal set. components of the relat ive wind.

T ( t h r u s t ) a x i s i s a l igned w i t h Hw (ho r i zon ta l ) a x i s is a l igned wi th the component of rela-

Yw ( s i d e f o r c e ) a x i s completes the The

This a x i s system i s shown in f i g u r e B-1 a long w i t h t h e

ROTOR SHAFT OR1 ENTATION

(a) PERSPECTIVE VIEW (b) PLAN VIEW NORMAL TO UH, VH PLANE

Figure B-1.- The hub-wind a x i s system wi th main r o t o r f o r c e , moment, and v e l o c i t y components def ined.

Hub Body

The hub-body system co inc ides wi th t h e hub-wind system when the s i d e s l i p ang le Bw i s zero . Thus, t he T a x i s i s a l igned wi th the s h a f t a x i s , and the f o r c e HR l i e s i n t h e XB - Z B p lane ( see f i g . B-2).

Body Center of Gravi ty

All f o r c e s and moments are expressed re la t ive t o the body-c.g. system f o r u s e i n t h e six-degrees-of-freedom r i g i d body equat ions of motion. This a x i s system has i t s o r i g i n a t the c e n t e r of g r a v i t y with the x a x i s a l igned wi th t h e l o n g i t u d i n a l a x i s of t h e h e l i c o p t e r , and t h e z a x i s ly ing on t h e p lane of symmetry ( see f i g . B-2).

1 3

HU B-BODY

'EM

AIRCRAFT REFERENCE SYSTEM

Figure B-2.- Hub-body, a i r c r a f t r e f e r e n c e and body-c.g. a x i s systems.

A i r c r a f t Reference

The a i r c r a f t r e f e rence axes are used t o l o c a t e a l l f o r c e and moment gene ra t ing components and t h e c e n t e r of g r a v i t y . The a i r c r a f t r e f e r e n c e axes are p a r a l l e l t o t h e body-c.g. axes. The a x i s o r i g i n i s t y p i c a l l y l o c a t e d ahead and below t h e a i r - c r a f t a t some a r b i t r a r y p o i n t w i th in t h e p l ane of s y m m e t r y . S t a t i o n s are measured p o s i t i v e a f t a long t h e l o n g i t u d i n a l a x i s . t o t h e p i l o t ' s r i g h t . Water l ines are measured v e r t i c a l l y , p o s i t i v e upward ( s e e f i g . B-2) .

B u t t l i n e s are l a t e r a l d i s t a n c e s , p o s i t i v e

Local Wind

.

The local-wind axes systems a r e employed f o r c a l c u l a t i o n of l i f t and d rag f o r c e s on t h e empennage and on t h e fuse l age . For each empennage s u r f a c e , t h e o r i g i n i s a t t h e q u a r t e r po in t of t h e mean aerodynamic chord, and t h e l i f t f o r c e i s normal t o t h e r e l a t i v e wind and t o a spanwise l i n e pas s ing through t h e q u a r t e r chord po in t .

.

14

APPENDIX C

MAIN ROTOR FLAPPING DYNAMICS AND FORCE AND MOMENT CALCULATION

The d e r i v a t i o n of t he t ip -pa th plane dynamic equat ion is given i n d e t a i l i n r e fe rence 2. For t h e non tee t e r ing N-bladed r o t o r , t h e t ip -pa th p lane dynamic equa- t i o n s are as fo l lows:

15

U n 3 m

+ W

I * r l lN + w 3 m

I 0 ?I* ffl

I

W * 10

21- I

N =I*

m + rl

I

@ N

m I 0

m

I 0 u

uXlc: [I)

3 wXlG

C I

- +

+ W

I

W rl I-

w 0 N Nlm I

N1l* Fl

+ v) 0 u n

"w (N c

W

n

31: 1w

II

8 2 +

+ + +

I I I I N C

II

tw

nll t!=l G + II

:ai ta

N

C

II

tY

16

where

yK1 +-+ - 8 K B P2 = 1 +-

IBn2 g-%

and

For a two-bladed t e e t e r i n g r o t o r , t h e t ip-path p lane r e p r e s e n t a t i o n l o s e s its phys ica l meaning. However, i f t h e approximation f o r b l ade f l app ing i s employed, t h a t is, 7

I* $ ( t ) = a o ( t ) - a l ( t ) c o s + - b l ( t ) s i n + then a. is t r e a t e d as a p r e s e t cons tan t . The c o e f f i c i e n t s a l ( t ) and b , ( t ) can then be solved f o r by s e t t i n g E = d o = S o = 0 i n equat ion (1) .

The f o r c e and moment express ions , with the harmonic terms dropped t h a t correspond t o t h e number of r o t o r b l ades are given below. given i n r e fe rence 2.

A d e r i v a t i o n of t h e s e equat ions i s

u2 2 *MR = nb 2 pacR(QR)2 + ( 1 - E ~ ) X + 8, [+ + 5 (1 - E)] + 0, [+ + - 4 ( 1 - E2)]

a - 1 (1 - E ~ ) ( B ~ ~ - K b ) - a. [$ +

+ - [k (1 -

(1 - E)]K~ + a, [; e ( l -

s i n B ) ] - - - (i - ;) 2 1 1

6 1 qH "bMB .. aO + 1 ( 1 - (? c o s Bw +

n 4 4 W g

17

c

18

n 7 4 - - (1 - ~ ) ~ a ~

- - 7 a (1 - E’)(- y PH s i n 6 + (% - a,)] 4 1 w R

19

H .

Q = nb pacR2 (RR) (-& [l + (1 - - ( e o - K l a o ) [$ + (5 - ') 4 Q + '(') 6 Q

- 4

u b l PH qH - - a. + - (1 - E 2 ) p 2 + + (- - s i n 6 + - c o s .)1 - K 1 1 a ) [(i - i ) (> + b,) :

6 16 R w n

-e For the case of a two-bladed t e e t e r i n g r o t o r , t h e f o r c e s and moments can be ..

obta ined by s e t t i n g E = 0 , io = a. = 0 i n the above equat ions . . The r o t o r p r o f i l e drag c o e f f i c i e n t 6 is r equ i r ed i n t h e computation of Hw and

An express ion f o r t h i s c o e f f i c i e n t w a s employed which provides a reasonable match Q. w i t h measured s e c t i o n c h a r a c t e r i s t i c s as follows:

6 = 0.009 + 0.3 (2s 20

where (6CT/aa) i s approximately equal t o an averaged equ iva len t r o t o r b l ade a n g l e of attack.

The r o t o r in f low r a t i o , def ined as

= - - WH cT

nR Z(p2 + X 2 ) l j 2

i s r equ i r ed i n t h e computation of t h rus t . t h e computer program through a Newton-Ralphson i terat ive technique.

This i m p l i c i t r e l a t i o n s h i p i s so lved i n

2

The main rotor-induced v e l o c i t y a t t h e d i s k i s r equ i r ed i n subsequent ca l cu la - t i o n s of t he rotor- induced v e l o c i t i e s on t h e h e l i c o p t e r components. The express ion

" f o r t h i s v e l o c i t y i s as fol lows:

Transformation T,

The c a l c u l a t i o n of t h e r o t o r f l app ing dynamics and t h e r o t o r f o r c e s and moments r e q u i r e s t h e angu la r v e l o c i t i e s and a c c e l e r a t i o n s expressed i n t h e hub-body system of axes and t h e a n g l e of s i d e s l i p a t t h e hub. I n a d d i t i o n , the c y c l i c p i t c h must b e expressed i n t h e hub-wind system. obta ined from those expressed i n the body-c.g. system as fol lows:

The r equ i r ed v e l o c i t i e s and a c c e l e r a t i o n s are

pH = p c o s is + r s i n is

6, = 6 c o s is + C s i n is

S u = tuB - rB(BL ) - qB(WLH - WL > ] c o s i H c .g . c .g .

S + + P B ( ~ ~ ~ . ~ , ) - qB(sTAc,g. - ST%)]sin i

'E-

- .

where is is t h e forward t i l t of the s h a f t axis re la t ive t o t h e body-c.g. Z-axis ( see f i g . B-2) and STAY WL, and BL are coord ina te s i n the a i r c r a f t r e f e rence system of axes . I t i s assumed t h a t t he r o t o r hub l i e s i n t h e BL = 0 p lane .

The s i d e s l i p of t h e r o t o r i s then def ined as:

21

(B, i n t h e hub-wind a x i s system. The express ions are:

is def ined as zero if VH = UH = 0.) F i n a l l y t h e c y c l i c p i t c h must b e expressed

A = Als c o s $ - B l s s i n Bw 1c W

s i n Bw + Bls COS Bw B l c =

Transformation T2

The r o t o r f o r c e s and moments c a l c u l a t e d i n the hub-wind axes system must be expressed i n t h e body-c.g. system as i n p u t s t o the six-degree-of-freedom rigid-body equat ions of motion. i n the hub-body sys t em of axes and then t r a n f e r r e d t o t h e body-c.g. system.

To accomplish this, t h e f o r c e s and moments are f i r s t expressed

The f o r c e s and moments i n the x-y p lane are modified by t h e f i r s t transforma- t i o n as fol lows:

HH = H c o s 8, + Yw s i n B, W

YH = -H s i n f3 + Y cos 6, W W W

% = Mw cos Bw + Lw s i n BW

= -Mw s i n €3 + L cos Bw LH W W

The t ransformation t o the body-c.g. system accounts f o r t he s h a f t - t i l t and t h e moment about t h e c.g. caused by the r o t o r forces . system are:

The f o r c e s expressed i n the body-c.g.

S = T s i n i - c o s i

S

Ym = YH

= -T c o s i - s i n i ‘MR S S

The moments i n t h e body-c.g. system are:

- ST+) - % ( m H - WL = I’$ - Zm(STA c .g . c .g .

= Q c o s i + L s i n is + % R ( ~ ~ c . g. NMR s H + YMR(STAc. g . - ST+)

22

APPENDIX D

TAIL ROTOR FLAPPING AND FORCE CALCULATION

The t a i l r o t o r is assumed t o b e t e e t e r i n g wi th a c o n s t a n t b u i l t - i n coning ao. The angu la r rate of t h e t a i l r o t o r i s s u f f i c i e n t l y h igh so t h a t t he t ip -pa th dynamics may b e ignored. assumed t o be cons t an t , and a s teady-s ta te s o l u t i o n of t h e equat ions i s obta ined t o y i e l d t h e s e q u a n t i t i e s as fol lows:

Under these condi t ions , a, and b, i n t h e f l a p p i n g equat ions are

-.

.

* .

where

4 2

ATR = 1 - - 4 TR + K:TR(l + F)(l + 5 PgR)

The f o r c e s on t h e t a i l r o t o r a r e then c a l c u l a t e d as fol lows:

- a OTR (i + *) KITR + 21

23

a blm 3 + - - 4

1 + jJ KlTRblTR -

'TR

The p o s i t i v e d i r e c t i o n s f o r t a i l r o t o r torque and r o t a t i o n are shown on f i g u r e D - 1 .

THRUST ALONG SHAFT INTO PAGE

TR

G ROTATION

Figure D-1. - Tail r o t o r f o r c e s and moments.

24

3 a o ~ b l ~ ~ 6

+ - A a - 4 TR 1TR

+ - + - ' ~ T R 3 ATR + alTRplTR] - i: 8 16 K 1 ~ ~ b l ~ ~ p m + 2 16 .

The r o t o r b l ade p r o f i l e drag c o e f f i c i e n t , 6 , is requ i r ed i n t h e computation of HTR and QTR. This i s c a l c u l a t e d s i m i l a r l y t o the case of t h e main r o t o r as fol lows:

3-

'.

The inf low r a t i o , as i n t h e c a s e of t he main r o t o r , must be c a l c u l a t e d from the fo l lowing r e l a t i o n s h i p :

25

This i m p l i c i t r e l a t i o n s h i p i s solved through an i t e r a t i o n procedure.

Main Rotor I n t e r f e r e n c e and Transformation T,

The c a l c u l a t i o n of the t a i l r o t o r f o r c e s and moments r e q u i r e s t h e l o c a l f low v e l o c i t y components i n t h e hub-wind axes system f o r t h e t a i l r o t o r . t h e t a i l r o t o r i nc ludes provis ion f o r t h e downwash con t r ibu ted by the ' f low f i e l d of t h e main r o t o r . For t h e s e c a l c u l a t i o n s , t h e t a i l r o t o r i s assumed t o l i e i n t h e x-z plane.

The v e l o c i t y a t

The v e l o c i t i e s a t t h e r o t o r hub i n an a x i s system co -d i r ec t iona l w i t h t h e body-c.g. system are:

B UTR = u

= vB + p B ( m m - WL 1 - rB(STATR - STA TR c .g . c.g.

wTR = w + qB(STATR - STA ) + wiTR B c.g.

The quan t i ty w i i s t h e downwash v e l o c i t y due t o t h e main r o t o r . The downwash x and s i d e s l i p . v e l o c i t y is r e p r e s e n t s i n t h e modes as a func t ion of t h e wake ang le

The v a r i a t i o n of the downwash v a r i e s non l inea r ly wi th l o c a t i o n i n t h e wake. Therefore , t h e downwash must be determined uniquely f o r each l o c a t i o n from d a t a such as those publ ished i n re ference 10. The r ep resen ta t ion i n the model c o n s i s t s of t h e downwash expressed as a func t ion of a power series i n the wake ang le a t ang le s of s i d e s l i p of 0", ?goo, and 180" as fol lows:

where v i i s t h e momentum-theory va lue of t h e rotor- induced v e l o c i t y . The coord ina te system employed i n r e fe rence 10 i s centered a t t h e hub, and t h e X and Y axes l i e i n the t ip -pa th plane. Accordingly, t h e t a i l r o t o r l o c a t i o n i n t h i s coord ina te system varies wi th the value of mate n a t u r e of t he wake estimate, which can vary widely wi th d i s k load d i s t r i b u t i o n , i n c l u s i o n of t h i s level of d e t a i l i s unwarranted. Accordingly, t h e downwash is esti- mated f o r a cons tan t l o c a t i o n re ferenced t o the hub-body system and t h e wake ang le r e f e r r e d t o t h a t sys tem. Hence,

a l , t he t ilt of t h e t ip -pa th plane. I n view of t h e approxi-

I n t e r p o l a t i o n f o r t he va lue of t he downwash a t a r b i t r a r y s i d e s l i p ang le s i s accom- p l i shed as i n t h e fo l lowing example, where B h a s been determined t o b e between 290": --*

Simi la r ly , when 90" < 6 270", t he downwash i s c a l c u l a t e d as fo l lows:

26

.

- If-

W iTR = [(?) lcos B I + (%) f3=9 0 I s i n B I ] V i i @ = l e 0

The advance r a t i o f o r t h e t a i l r o t o r , which is assumed t o be i n the x-z p lane , i s then:

and the r o t o r s i d e s l i p ang le i s def ined f o r t h e t a i l r o t o r as:

Then, i n t h e hub-wind a x i s system defined f o r t h e t a i l r o t o r , the angular v e l o c i t i e s are :

Transformation T,

The t ransformat ion t o express t h e t a i l r o t o r f o r c e s and moments i n t h e body-c.g. system of axes i s as fo l lows:

XTR = - ( Y T R ) ~ s i n BTR - (HTR)~ cos BTR

27

APPENDIX E

EMPENNAGE FORCES AND MOMENTS

The express ions f o r t h e l i f t and d rag c o e f f i c i e n t s f o r t h e vertical f i n and t h e h o r i z o n t a l s t a b i l i z e r are i d e n t i c a l when re ferenced t o a l o c a l wind-axes system f o r t h e p a r t i c u l a r sur face . (See appendix B.)

The cond i t ions a t t h e s t a l l of each su r face are def ined as fol lows:

IT as 1. - cLM as - - a

IT as = - 4

a1 = 1.2as

IT a s - 4

71 < - a s - 4

The ang le of a t t a c k f o r i npu t i n t o the express ions f o r t h e l i f t and drag c o e f f i c i e n t is:

IT 0 5 a < - 2 ai = a

IT - - - I a < o 2 ai = -a

IT - T I I a < - -

2 a i = 7 1 + a

where ai r ep resen t s t h e angle-of-at tack input t o t h e l i f t and drag equat ions .

The l i f t and drag c o e f f i c i e n t s are c a l c u l a t e d us ing t h e fo l lowing express ions :

0 5 a i < as cLo = aai

IT = -0.1254 + 0.9415ai + 0.977525 s i n 2 a i 0.35 < a i < 7 cDP

28

where C D ~ is t h e p r o f i l e drag c o e f f i c i e n t .

c; cD = ‘DP + 0.8rAR

Depending on t h e quadrant i n which the ang le of a t t a c k l i es , t h e l i f t c o e f f i c i e n t f o r t h e s u r f a c e is:

n O Z a < - CL = CL, 2

?T CL = -CLo - - < a < I T

CL = -0.8CL IT

2 -

IT

0 2

71 - . r r _ < a < - - 2

CL = 0.8C~ a

Horizonta l T a i l

The l i f t curve s lope of t he ho r i zon ta l t a i l i s c a l c u l a t e d from elementary l i f t - i n g l i n e theory and is co r rec t ed f o r s i d e s l i p as fo l lows:

Main r o t o r i n t e r f e r e n c e and t ransformation T5- The v e l o c i t y a t the h o r i z o n t a l t a i l i n c l u d e s a f a c t o r t o account for r o t o r downwash, W i H T .

l a t e d s i m i l a r l y t o t h a t c a l c u l a t e d f o r t h e t a i l r o t o r . I n c a l c u l a t i n g the v e l o c i t i e s a t t h e h o r i z o n t a l t a i l , t h e r e l a t i v e l y small c o n t r i b u t i o n of r o l l rate is ignored. The expres s ions f o r t h e v e l o c i t y components a t t h e h o r i z o n t a l t a i l are as fol lows:

This f a c t o r i s ca lcu-

The a n g l e of a t t a c k and ang le of s i d e s l i p of t h e h o r i z o n t a l t a i l are then:

’H S = sin-’ cz) where

29

The a n g l e iHs cambered s u r f a c e may be accounted f o r by s e l e c t i n g t h i s parameter t o d i f f e r from t h e ang le of t h e chord p l ane by an amount equal t o t h e ang le of attack f o r zero l i f t .

i s t h e inc idence of t h e h o r i z o n t a l t a i l . Note t h a t t h e e f f e c t of a

The f a c t o r employed t o c a l c u l a t e l i f t and drag f o r c e s from t h e l i f t and drag c o e f f i c i e n t s is:

where t h e cons t an t ~ H S t a i l area considered t o be w i t h i n t h e fuse lage .

i s used t o account f o r fuse l age blockage o r t h e h o r i z o n t a l

Transformation T6- The f o r c e s i n the body-c.g. system of axes a t t r i b u t e d t o t h e h o r i z o n t a l t a i l are:

The f o r c e s a r e assumed t o a c t a t the q u a r t e r chord p o i n t of t he h o r i z o n t a l t a i l a t a spanwise l o c a t i o n corresponding t o the c e n t e r of area. The moments are:

Vertical F in

The estimate of t h e l i f t curve s lope of t h e v e r t i c a l f i n inc ludes t h e e f f e c t of sweep as w e l l as of s i d e s l i p as fol lows:

Transformation T7- The computation of t he v e l o c i t y a t t h e v e r t i c a l f i n ignores t h e s m a l l con t r ibu t ion from r o l l ra te and inc ludes a t e r m i n t h e sidewash due t o t h e induced v e l o c i t y of the t a i l r o t o r . The t h r e e components of v e l o c i t y are as fo l lows:

30

The cons t an t immersed i n t h e t a i l r o t o r wake. The main r o t o r i n t e r f e r e n c e f a c t o r wi is computed

s i m i l a r l y t o t h e f a c t o r f o r downwash a t t h e t a i l r o t o r .

kvTR can be ad jus t ed t o account f o r t h e f r a c t i o n of t h e v e r t i c a l t a i l

F

I The ang le s of a t t a c k and s i d e s l i p of t h e v e r t i c a l f i n i n t h e l o c a l wind coord i - n a t e system are:

.-

where

As i n t h e case of t h e h o r i z o n t a l t a i l , t h e ang le camber of t h e v e r t i c a l f i n .

iF can be ad jus t ed t o account f o r

31

APPENDIX F

CALCULATION OF FUSELAGE FORCES AND MOME"l'S

I n c a l c u l a t i n g the fuse l age f o r c e s and moments, i t i s assumed that t h e long i tud i - n a l f o r c e s and moments are dependent on a n g l e of a t t a c k and t h e lateral f o r c e s and moments are dependent on ang le of s i d e s l i p . i s assumed t o have a c o n t r i b u t i o n from bo th angle of a t t a c k and s i d e s l i p . r e s e n t a t i o n i s a gross s i m p l i f i c a t i o n , and i f d e t a i l e d test d a t a are a v a i l a b l e on

e .g . , r e f . 8 . )

The except ion is t h e d rag f o r c e , which This rep-

which t o base a more s o p h i s t i c a t e d r e p r e s e n t a t i o n , they should be s u b s t i t u t e d . (See, I_

Low-angle approximation a,B = -15" t o 15"

D = D, + D B

High-angle approximation

a 2 D / q D B = qf ( aB2 B2)

s i n a1 s in2 , )

s i n B cos B ) s i n B 1

M = qf (M,=90~ I s in a I s i n a) N = qf ( ~ ~ = ~ ~ 0 I s i n B I s i n B)

D = D, + DO DB = qf(Dg=go" I s i n B I s i n 6') - .

Phasing between the low-angle approximation and t h e high-angle approximation i s The complex phase a n g l e i s based on a complex phase ang le and l i n e a r i n t e r p o l a t i o n .

given by :

Linear i n t e r p o l a t i o n based on t h i s phase a n g l e i s employed over t h e range of 15" < ITFUSI 30".

32

Main r o t o r i n t e r f e r e n c e and t ransformat ion Tg- The f u s e l a g e ang le s of a t t a c k , a n g l e of s i d e s l i p , and dynamic p res su re inc lude an averaged e f f e c t of r o t o r downwash. The express ion f o r downwash w a s ob ta ined f o r t y p i c a l s i n g l e r o t o r h e l i c o p t e r s from an empi r i ca l f i t t o d a t a presented i n r e fe rence 10. are as fo l lows:

The v e l o c i t i e s on the fuse l age

f wf = WB + w i .-

Where wif i s t h e induced v e l o c i t y due t o the r o t o r and is def ined as:

(F) = 1.299 + 0 . 6 7 1 ~ - 1 . 1 7 2 ~ ~ + 0 . 3 5 1 ~ ~

The r o t o r wake angle x i s def ined as:

The a n g l e s of a t t a c k and s i d e s l i p a r e then:

Wf af = tan-’ - Uf

where

The dynamic p res su re on t h e fuse l age i s based on the v e l o c i t y , inc luding the e f f e c t ’- of r o t o r downwash.

Transformation T lo - The fuse l age f o r c e s and moments are c a l c u l a t e d i n the wind system of axes about t h e fuse l age re ference po in t . The t ransformat ion must t r a n s f e r t h e moment t o t h e c e n t e r of g r a v i t y and express a l l f o r c e s and moments i n the body system of axes.

33

The forces and moments in the body-c.g. system of axes are given by the follow- ing equations:

Xf = -D cos B cos a - Y sin f3 sin a + L s in a

Yf = Y cos B - D s in B

Zf = -L cos a - D cos B s in a - Y s in s in a

34

RPM GOVERNOR

c

.

The rpm governor model provides f o r an rpm degree of freedom w i t h simple engine and governor dynamics. When the rpm governor op t ion is used, t h e main r o t o r and t a i l r o t o r speeds are changed based on c u r r e n t to rque requirements and engine power avail- ab le . torque. ments which cause r o t o r speed v a r i a t i o n s which feed through t h e governor c o n t r o l l a w s t o provide f u e l f low changes. which are converted t o engine torque ava i l ab le .

T r i m i n i t i a l c o n d i t i o n s e s t a b l i s h b a s e l i n e va lues of r o t o r speed and engine F l i g h t v a r i a t i o n s from t h a t t r i m cond i t ion r e s u l t i n changing torque requi re -

The f u e l f low changes provide engine power changes

The rpm degree of freedom may b e descr ibed by the equat ion:

Fo

where :

QE = engine torque

QR = r equ i r ed r o t o r to rque

J = r o t o r r o t a t i o n a l i n e r t i a

h = r o t o r speed ra te of change

low f l a p p i n g hinge o f f s e t s , t h e r o t t h e b l a d e f l a p p i n g i n e r t i a , IB:

t i o n a l i e r t i , J , may be a p roximated from

J = NIB

where N is t h e number of b lades .

The main - \ ' t o r to rque requi red i s a complicated func t ion of many v a r i a b l e s i n c l u d i n g b l ade p i t c h s e t t i n g s , a i r speed , in f low v e l o c i t y , f l a p p i n g ang le s , and r o t o r speed. The torque equat ion provides the necessary c a l c u l a t i o n . The r o t o r speed gov- e r n i n g of t h i s model u ses l i n e a r c o n t r o l theory based on an ope ra t ing po in t . Thus, t h e r o t o r speed, Q, t akes the form:

R = R, + AR

where R, i s the trim r o t o r speed and AR i s t h e rotor-speed v a r i a t i o n . ?*

A d e t a i l e d r e p r e s e n t a t i o n of engine torque r e q u i r e s a complicated non l inea r f u n c t i o n of many v a r i a b l e s inc lud ing opera t ing power s e t t i n g , ambient p re s su re and

f u e l f low t o c o n t r o l engine torque. be conver ted t o torque based on t h e c u r r e n t r o t o r speed.

temperature , and f u e l f low, W f . For t h i s s i m p l i f i e d model, rpm governing acts on A gas t u r b i n e engine produces power which must

35

It i s important t o n o t e that t h e c u r r e n t r o t o r speed is used, n o t t h e ope ra t ing po in t speed, Q,,. Thus, the power-to-torque conversion f a c t o r is always changing.

The engine power response t o f u e l f low i s simply modeled as a f i r s t o rder lag:

The time cons tan t , T E , and ga in , KE, are s e l e c t e d based on engine c h a r a c t e r i s t i c s and opera t ing poin t . -

The complete rpm degree of freedom and engine-governor model is shown i n f i g u r e 3.

Note t h a t a t h r o t t l e on/off switch has been added t o a l low tu rn ing t h e power on o r o f f . power change time cons t an t f o r t he t h r o t t l e and a small power change t i m e cons t an t f o r normal engine opera t ion .

A sepa ra t e time cons t an t f o r t h e t h r o t t l e i s provided t o a l low f o r a l a r g e

The governor c o n t r o l l a w is given by:

A W E = -@g1 +

which provides p ropor t iona l , i n t e g r a l , and rate feedback.

-s

.

36

APPENDIX H

COCKPIT CONTROLS AND CYCLIC CONTROL PHASING

Y

Cycl ic , c o l l e c t i v e , and pedal c o n t r o l s may be s p e c i f i e d from a ze ro p o s i t i o n e i t h e r cen te red o r f u l l l e f t o r down. between l o n g i t u d i n a l and la teral c y c l i c , t h a t is, cen te red o r f u l l l e f t and forward. The ze ro p o s i t i o n of t he c o l l e c t i v e o r peda l s may be s p e c i f i e d independently. The t a b l e below lists t h e zero p o s i t i o n , p o s i t i v e d i r e c t i o n , and s i g n of t h e moment or f o r c e produced i n t h e body system of axes. t i o n s are l i s t e d i n parentheses .

The c y c l i c c o n t r o l p o s i t i o n must be c o n s i s t e n t

A l t e r n a t e c o n t r o l zero p o s i t i o n conven-

Control

Longi tudina l c y c l i c , 6,

Lateral c y c l i c , 6,

C o l l e c t i v e , tic

Peda l s , 6p

Zero p o s i t i o n P o s i t i v e d i r e c t i o n Moment/force

Centered Af t ( f u l l forward)

Centered Right ( f u l l lef t )

+M

+L

F u l l down UP -Z

Centered Right pedal +N ( f u l l l e f t ) forward

The c o n t r o l gear ing , r i gg ing , and c y c l i c c o n t r o l phasing are governed by t h e follow- i n g equa t ions :

e, = C,6, + c,

= C b + c , 'OTR 8 P

T h e terms , C , , and C, are the cons t an t o r r i gg ing terms f o r each

c o n t r o l . The terms C, through C, can be used t o a d j u s t f o r t h e phase ang le between 2- 4 t h e c y c l i c c o n t r o l i npu t and the r e s u l t i n g f lapping . These phase ang le terms may b e

s p e c i f i e d i n two ways. on t h e b a s i s of main r o t o r dynamic p rope r t i e s :

F i r s t , a c y c l i c c o n t r o l phase r e l a t i o n s h i p may be s p e c i f i e d c

(CK2) 1 - ( 8 / 3 ) ~ + 2~~

1 - ( 4 / 3 ) € c, =

37

c, = ‘C, (s) Y

The terms CK1 and CK2 a r e the long i tud ina l and l a t e r a l c y c l i c c o n t r o l s e n s i t i v i t i e s , r e spec t ive ly . provides the fol lowing r e l a t i onsh ips :

As an a l t e r n a t i v e , a c o n t r o l phase angle , J l o , may be s p e c i f i e d , which

C, = (CK2)cos q 0 .

C, = (CK2)sin Jl,,

c, = -c, (s)

.

38

APPENDIX I

LINEARIZED SIX-DEGREE-OF-FREEDOM REPRESENTATION OF HELICOPTER DYNAMICS

The linear , f i r s t - o r d e r set of d i f f e r e n t i a l equa t ions desc r ib ing the r i g i d body motion of t h e h e l i c o p t e r are of the form:

2 = [FIX + [GI6

yr where x r e p r e s e n t s the p e r t u r b a t i o n s from t r i m of the state variables UB, WB, qBy 0 , VB, PB, $I, and r B ; and 6 r ep resen t s t h e d e v i a t i o n s from t h e t r i m c o n t r o l posi- t i o n s of 6,, 6,, 6,, and 6p.

=- angu la r v e l o c i t i e s p ~ , qBy and r B are zero. The l i n e a r r e p r e s e n t a t i o n i s v a l i d only i f the i n i t i a l

The elements of t h e F and G matrices are of two types. The f i r s t type c o n s i s t s of i n e r t i a l and g r a v i t a t i o n a l terms t h a t can be obtained a n a l y t i c a l l y from t h e equa- t i o n s of motion. The second type c o n s i s t s of p a r t i a l d e r i v a t i v e s a r i s i n g from aero- dynamic f o r c e s and moments. e r i n g both p o s i t i o n and nega t ive pe r tu rba t ions from t r i m .

The f o r c e and moment d e r i v a t i v e s are obta ined by consid- For example:

The elements of t h e F and G ma t r i ces and t h e s t a t e v a r i a b l e v e c t o r s are given i n t a b l e 1-1.

39

a

a

m a

ma a

a

m a

8

a

0

0.

3" a m a v

E+ n

W 4 a

m W a

U W a

" W

4 v

II II

x w

I I I I I I I

I I I I I I I

h

N N X Y

I

H

H

II

H

N

X v

U

I I

I I

I I

7 - - r - - - _ - I I

I I

I I

I I

40

APPENDIX J

CONFIGURATION D E S C R I P T I O N REQUIREMENTS

Table J-1 l ists t h e parameters requi red t o desc r ibe a h e l i c o p t e r conf igu ra t ion f o r u se i n t h e computer s imula t ion . L i s t ed are t h e parameter name, a l g e b r a i c symbol, computer mnemonic, and u n i t s f o r each parameter. In a d d i t i o n , a n example va lue based on an AH-1G is provided f o r each parameter. r e f e r e n c e 8.

AH-1G parameters are based on va lues i n

8 .

c

2-

4 1

TABLE J-1.- CONFIGURATION DESCRIPTION REQUIREMENTS

Units Example va lue

Algebraic CmPute r symbol mnemonic

Name

Main r o t o r (MR) group

MR r o t o r r a d i u s

MR chord

MR r o t a t i o n a l speed

Number of b l a d e s

MR Lock number

MR hinge o f f s e t

MR f l app ing s p r i n g c o n s t a n t

MR p i t ch - f l ap coupling tangent of

MR b l ade t w i s t

MR precone a n g l e ( r equ i r ed f o r t e e t e r i n g r o t o r )

MR s o l i d i t y

MR l i f t curve s lope

MR maximum t h r u s t

MR l o n g i t u d i n a l s h a f t t i l t ( p o s i t i v e forward)

MR hub s t a t i o n l i n e

MR hub w a t e r l i n e

T a i l r o t o r (TR) group

TR r a d i u s

TR r o t a t i o n a l speed

TR Lock number

TR s o l i d i t y

TR p i t ch - f l ap coupling tangent of

TR precone

TR b l ade t w i s t

TR l i f t curve s lope

TR hub s t a t i o n l i n e

TR hub w a t e r l i n e

6 3

6 3

RMR

QMR

YMR

KB

K l

nb

E:

‘ ~ M R

OMR a

UMR

aMR

‘Tmax iS

STAH

WLH

RTR

QTR

YTR

OTR

K 1 ~ ~

OTR

‘ ~ T R a~~

WLTR

a

STAm

ROTOR

CHORD

OMEGA

BLADES

GAMMA

EPSLN

AKBETA AKONE

f t

ft

r ad / sec

N-D N-D

percent/100

lb-f t / r a d

N-D

THETT r ad

AOP rad

SIGMA N-D

ASLOPE rad’’

CTM N-D

CIS r ad

S TAH i n .

WLH i n .

RTR f t

OMTR r ad 1 sec

GAMATR N-D

STR N-D

FKITR N-D

AOTR r ad

THETR r a d

ATR rad’’

STATR i n .

WLTR i n .

22

2.25

32.88

2 m

5.216

0

0

0

-

-0.17453

0.048

0.0651

6.28

0.165

0

200

152.76

4 .25

168.44

2.2337

0.105

0.5773

-9 0.02618

0

6.28

520.7

118.27

42

TABLE J-1. - Continued . Name A1 g eb ra i c C ompu t e r Examp l e Uni t s symbol mnemonic va lue

A i r c r a f t mass and i n e r t i a

A i r c r a f t weight

A i r c r a f t roll i n e r t i a

A i r c r a f t p i t c h i n e r t i a

A i r c r a f t yaw i n e r t i a

A i r c r a f t c r o s s product of i n e r t i a

Center of g r a v i t y s t a t i o n l i n e

Center of g r a v i t y w a t e r l i n e

Center of g r a v i t y b u t t l i n e

Fuselage (Fus)

Fus aerodynamic r e fe rence p o i n t s ta t ion l i n e

Fus aerodynamic r e fe rence p o i n t wa t e r 1 i n e

Fus drag , c1 = B = 0

Fus drag , v a r i a t i o n w i t h ci

Fus drag , v a r i a t i o n wi th a2

Fus d rag , v a r i a t i o n wi th B 2

Fus drag , cx = 90"

Fus drag , 6 = 90"

Fus l i f t , a = B = 0

WAITIC

X I X X I C

X I Y Y I C

X I Z Z I C

X I X Z I C

STACG

WLCG

BLCG

STAACF

WLACF

D 1

D2

D3

D4

D5

D6

l b

slug-f t

slug-f t

slug-f t

slug-f t

i n .

i n .

i n .

i n .

i n .

f t2

f t 2 / r a d

f t / r ad2

f t 2 / r a d 2

f t 2

f t2

XLO f t 2 LO - q

8000

2700

12800

10800

950

196

73

0

200

54

5 .5

-4.01

41.56

141.16

84.7

156.1

-4.11

XL1 f t 2 / r a d 15.64 Fus l i f t , v a r i a t i o n wi th a

Y 1 f t 2 / r a d 93.85 Fus s i d e f o r c e , v a r i a t i o n wi th B

B at3 Y L 1 f t 3 / r a d 246.31 Fus r o l l i n g moment, v a r i a t i o n wi th a ( R / q )

a (L/q) aa

a ( Y / q ) a 6

Fus r o l l i n g moment, B = 90" YL2 f t 3 0

43

TABLE J-1.- Continued.

Algebraic Computer Example va lue N a m e symbol mnemonic Un i t s

Fus p i t c h moment, a = B = 0

Fus p i t c h moment, v a r i a t i o n wi th a

Fus p i t c h moment, a = 90”

Fus yaw moment, v a r i a t i o n wi th f3

Fus yaw moment, B = 90”

Horizonta l s t a b i l i z e r (HS)

HS s t a t i o n

HS w a t e r l i n e

HS inc idence angle

HS area

HS aspect r a t i o

HS maximum l i f t curve s l o p e

HS dynamic pressure ratio

Main r o t o r induced v e l o c i t y e f f e c t a t HS

Vertical f i n (VF)

VF s ta t i o n l i n e

V e r t i c a l f i n wa te r l ine

VF Incidence angle

VF area

VF aspect r a t i o

VF sweep ang le

VF maximum l i f t curve s lope

VF dynamic pressure r a t i o

T a i l r o t o r induced v e l o c i t y e f f e c t a t VF

M 9

a (M/q) aa

MI 4 a=90

a ( N / q )

STAHS

WLHS

iHS

sHS

ARH S

C

“HS

ImaxHS

K~~~

STAVF

WLVF

iVF

SVF

ARVF

AF

ImaxVF C

qVF

k~~~

XM1

x M 2

xM3

xN1

xN2

STAHS

WLHS

AIHS

SHS

ARHS

CLMHS

X”

XKVMR

STAVF

WLVF

AIFF

SF

AFtF

ALMF

CLMF

VNF

XKVTR

f t j -6.901

f t3 / rad 280.405

m f t 3 300

f t3 / rad -913.35

f t 3 -600

i n .

i n .

rad

f t 2

N-D

N-D

N-D

N-D

i n .

i n .

r ad

f t 2

N-D

rad

N-D

N-D

N-D

398.5

56.0 0

14.7

3.0 1 .2

0.8 1.0

50 1

84 0

18.6

1.56 0.7853

1.2

0.9 0.9

-?

s c

44

TABLE J-1.- Concluded.

N a m e Algebraic Computer Uni t s Example symbol mnemonic va lue

Cont ro ls

ze ro la teral c y c l i c s t i c k Swashplate lateral c y c l i c p i t c h f o r C A ~ ~ CAI S rad 0.0

Swashplate l o n g i t u d i n a l c y c l i c p i t c h CB1 CBIS rad & f o r ze ro l o n g i t u d i n a l c y c l i c s t i c k S c

0.0

Longi tudina l c y c l i c c o n t r o l C K l CK1 r ad / in . 0.03927

Lateral c y c l i c c o n t r o l s e n s i t i v i t y CK, CK2 r a d / i n . 0.02618

Main r o t o r r o o t c o l l e c t i v e p i t c h f o r C, c5 rad 0.1501 ze ro c o l l e c t i v e s t i c k

Main r o t o r c o l l e c t i v e c o n t r o l C6 C6 r a d / i n . 0.036652 s e n s i t i v i t y

b s e n s i t i v i t y +

T a i l r o t o r r o o t c o l l e c t i v e p i t c h f o r C, c7 rad ze ro pedal p o s i t i o n

0.11781

Pedal s e n s i t i v i t y ‘8 C8 r a d / i n . 0.08055

45

REFERENCES

l . 1 . S i n a c o r i , J. B. ; S t ap le fo rd , R. L.; Jewell, W. F.; and Lehman, J. M.: Researcher 's Guide t o the NASA Ames F l i g h t Simulator f o r Advanced A i r c r a f t (FSAA), NASA CR-2875, Aug. 1977.

2. Chen, R. T. N.: A Simpl i f ied Rotor System Mathematical Model f a r P i l o t e d F l i g h t Dynamics Simulat ion, NASA TM-78575, May 1979.

' 3. Chen, R. T. N.: E f f e c t s of Primary Rotor Parameters on Flapping Dynamics, NASA TP-1431, Jan. 1980. Y -

4. Gessow, A.; and Meyers, Jr., G. D.: Aerodynamics of t h e Hel icopter . F rede r i ck L Ungar Pub. Co. (New York), 1952. A

5. Seckel , E . ; and C u r t i s s , Jr., H. C.: Aerodynamic C h a r a c t e r i s t i c s of He l i cop te r Rotors . Dept. of Aerospace and Mechanical Eng. Rep. 659, P r ince ton Univ., 196 2.

6 . White, F.; and Blake, B. B.: Improved Method of P r e d i c t i n g He l i cop te r Con t ro l Response and Gust S e n s i t i v i t y . Paper 79-25, 35th Ann. N a t . Forum of t h e AHS, Wash., D.C . , May 1979.

7. Chen, R. T. N. : S e l e c t i o n of Some Rotor Parameters t o Reduce Pi tch-Roll Coupling of He l i cop te r F l i g h t Dynamics. Na t iona l S p e c i a l i s t ' s Meeting on Rotor System Design, P h i l a d e l p h i a , Penn.,

P r e p r i n t No. 1-6, paper presented a t t h e AHS

Oct. 22-24, 1980.

8. Davis, J. M . ; Bennet t , R. L. ; and Blankenship, B. L.: R o t o r c r a f t F l i g h t Simula- t i o n wi th A e r o e l a s t i c Rotor and Improved Aerodynamic Representa t ion , Vol. 1 - Engineer ' s Manual, USAAMRDL-TR-74-1OA, June 1974.

9. M i l i t a r y S p e c i f i c a t i o n , F ly ing Q u a l i t i e s of P i l o t e d Airp lanes , MIL-F-8785B(ASG), Aug. 1969.

10. Jewel, Jr . , J. W . ; and Heyson, H. H.: Cha r t s of Induced V e l o c i t i e s Near a L i f t - i n g Rotor , NASA MEMO 4-15-59LY May 1959.

-?

46

b r

1. Report No. NASA TM-84281

8 ,,

2. Govmmrnt Acclrion No. 3. Rucipient's Catalog No.

4. Title and Subtitle

SeDtember 1982 6. Performing Orgniration codr

A MATHEMATICAL MODEL OF A SINGLE MAIN ROTOR HELICOPTER FOR PILOTED SIMULATION

5. Report Date

7. Author(s1 P e t e r D. Ta lbot , Bruce E. T in l ing , W i l l i a m A. Decker, and Robert T. N . Chen

9. Performing Organization Name and Address

NASA Ames Research Center Moffe t t F i e l d , C a l i f o r n i a 94035

8. Performing Organization Report No.

A-9033 10. Work Unit No.

T-6292Y 11. Contract or Grant No. I

. 12. Sponsoring Agency Name and Address

Nat iona l Aeronaut ics and Space Adminis t ra t ion Washington, D.C. 20546

13. Type of Report and Pari4 Cotmod

Technical Memorandum 14. Sponrorir) Agency Code

15. Supplementary Notes

7. Key Words 6uggestd by Authorlsl) Hel i cop te r F l i g h t s imula t ion ; Handling q u a l i t i e s H e l i c o p t e r aerodynamics He l i cop te r f l i g h t dynamics H e l i c o p t e r s t a b i l i t y and c o n t r o l aug-

menta t ion sy s t e m

Po in t of Contact: W i l l i a m A. Decker, Mail S top 211-2, NASA Ames Research Center , Moffe t t F i e l d , CA 94035 (415) 965-5362 o r FTS 448-5362

18. Distribution Statement

Unlimited

Subjec t Category - 08

~~ ~~

16. Abstract

This r e p o r t documents a h e l i c o p t e r mathematical model s u i t a b l e f o r p i l o t e d s imula t ion of f l y i n g q u a l i t i e s . The mathematical model is a non- l i n e a r , t o t a l f o r c e and moment model of a s i n g l e main r o t o r h e l i c o p t e r . The model h a s t e n degrees of freedom: s i x rigid-body, t h r e e ro to r - f l app ing , and t h e r o t o r r o t a t i o n a l degrees of freedom. The r o t o r model assumes r i g i d b l a d e s w i t h r o t o r f o r c e s and moments r a d i a l l y i n t e g r a t e d and summed about the azimuth. The f u s e l a g e aerodynamic model u ses a d e t a i l e d r e p r e s e n t a t i o n over a nominal ang le of a t t a c k and s i d e s l i p range of +15", and i t u s e s a s i m p l i f i e d curve f i t a t l a r g e angles of a t t a c k o r s i d e s l i p . S t a b i l i z i n g s u r f a c e aerodynamics are modeled with a l i f t curve s lope between stall l i m i t s and a gene ra l curve f i t f o r l a r g e a n g l e s of a t t a c k . A gene ra l i zed s t a b i l i t y and c o n t r o l augmentation system is descr ibed . Add i t iona l computer sub rou t ines provide op t ions f o r a s i m p l i f i e d engine/governor model, atmo- p h e r i c tu rbulence , and a l i n e a r i z e d six-degree-of-freedom dynamic model f o r s t a b i l i t y and c o n t r o l a n a l y s i s .

U n c l a s s i f i e d I 55 I A04 Unc las s i f i ed

*For =le by the National Technical Information Stvice. Springfield, Virginia 22161

L